I haven't had time yet to look into the phenomenon in the Hellenistic world.
But, if I may make a suggestion, it seems to me that a more appropriate
place to look for the putative background of the phenomenon as it occurs in
the NT is not the Hellenistic world at all. Instead, perhaps an examination
of the Pseudepigraphal material would be more enlightening.

In short, the NT authors seem much more influenced by thier jewish roots
than they do by the hellenistic "World View". In Judaism in the first c.
CE, angels were a source of great discussion. Perhaps, then, the phrase in
question sprung from that rather fertile ground rather than Hellenisms
stagnant soil.

So, in 1 Enoch 48, and 90ff we have extensive discussions of angels. In
Charlesworth's edition of the OT Pseud., under Glossalalia in the index he
has, in parentheses (language of angels) and he points to: the phenomenon in
Acts, in Merkabah mysticism, and in the Testament of Job. This, it seems to
me, is the place to start for an understanding of the phenomenon.